7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Numerical Study on Surface Tension Effects on Cavitation Bubble Collapse in Spherical
Acoustic and Nonspherical Rayleigh collapse
Ehsan Samiei1, Mehrzad Shams1, Miralam Mahdi1, Reza Ebrahimi2, and GoodarzAhmadi3
'Dep. Mechanical Engng., K.N. Toosi University of Tech.
Mollasadra St., Tehran, 193951999, Iran
e.samiei@yahoo.com, shams@kntu.ac.ir, miralammahdi@yahoo.com
2 Dep. Aerospace Engng., K.N. Toosi University of Tech.
Vafadar St., Tehran, 167653381, Iran
rebrahimi@ckntu.ac.ir
3Dep. Mech. & Aero. Enging., Clarkson University
Potsdam, NY, 13699, USA
ahmadi@clarkson.edu
Keywords: Acoustic and Rayleigh bubble collapse, surface tension, Gilmore Equation, VOF method.
Abstract
In the present study surface tension effect on cavitation bubble collapse is studied in ultrasound single spherical bubble
collapse and Rayleigh collapse of a nonspherical bubble near a rigid boundary. Gilmore equation is used to simulate spherical
bubble dynamics, with considering mass diffusion and heat transfer. In the nonspherical case the governing equations are
NavierStockes and energy. Ideal gas is assumed for bubble contents, and VOF method is used to track the interface. In the
case of Rayleigh collapse, the collapse stage is simulated in different conditions of bubble radius, and pressure difference
between bubble contents and surrounding liquid. Results show that surface tension does not have a significant role in collapse
rate and jet pattern. However, this effect is more important for smaller bubbles and lower values of pressure difference
between bubble contents and surrounding liquid. Increasing surface tension coefficient results in a small increment in jet
velocity. Similar results are obtained for spherical cases, but the increment in bubble surface velocity is higher than
nonspherical case. Considering both growth and collapse stages in spherical case shows that surface tension has an important
effect on growth stage. Lower values of surface tension result in larger maximum radius and longer growth duration, and more
violent collapse.
Introduction
Cavitation is a hydrodynamic phenomenon which is known
as one of the most important causes of erosion in hydraulic
devices. In some parts of the flow which the local pressure
drops down the vapour pressure of the fluid at the local
temperature, due to evaporation of liquid into the micro
bubbles carrying in the flow this micro bubbles start to grow
and when they are carried to the high pressure parts of the
flow, they collapse (Brennen 1995). If the collapse process
happens close enough to a rigid boundary, the bubbles shape
will not remain spherical and a jet forms in the part of the
bubble which is far from the rigid boundary and its direction
is toward the wall (Naude & Ellis 1961). In acoustic
cavitation which can happen in quiescent water, as a result
of the contact of a shock wave with a micro bubble, the
bubble grows and collapses (Brennen 1995).
Since this phenomenon was introduced by Reynolds (1873),
a large number of researches have been done to understand
the mechanism of erosion and the dynamics of cavitation
bubbles in order to either reduce the erosion rate in
hydraulic machinery or use cavitation in some medical and
industrial applications. Rayleigh (1917) developed a one
dimensional model for simulation of cavitation bubble
dynamics without counting the effects of surface tension
and mass transfer, and concluded the erosion in materials is
due to resultant high pressure around the bubble at the
collapse moment. Later, this model was developed by
Plesset (1949) assuming adiabatic dynamics of the bubble.
Plesset & Zwick (1952) showed that neglecting heat
transfer from the bubble to the liquid affects the dynamics
of bubbles significantly and developed a model for
calculation of heat transfer in the condition which the
thickness of thermal boundary layer is much smaller that the
bubble radius. This effect was developed later by
Prosperetti (1977), Plesset and Prosperetti (1997), and
Kamath et al. (1993). Herring (1941) showed that
compressibility effect is important in the latest moments of
collapse stage and implemented it in his model. This model
was developed by Gilmore (1952). Mass transfer effects
were showed to have important roles in collapse rate and
were implemented by Toegel et al. (2000) and Holzfuss
(2005).
Since Naude & Ellis (1961) introduced jet formation during
bubble collapse near a rigid boundary using high speed
photography, erosion and noise in hydraulic machinery has
been imputed to jet impact to the wall or the water hammer
generated due to jet impact to the other side of the bubble.
Philip and Lauterborn (1998) studied the dynamics of
growing and collapsing bubbles in different distances from
the wall and concluded that water hammer pressure of the
bubble which are closer to the wall is more significant.
Lindau and Lauterborn (2003) investigated that by collapse
of the bubbles with small values of standoff (the ratio
between the distance of bubble centre to the wall and radius
of the bubble, y) the bubble pattern becomes toroidal, and if
the standoff is large enough, due to the shock wave
generated by jet impact to the other side of the bubble a
counter jet forms after the collapse stage. Popinet and
Zaleski (2002) studied on viscous effects on cavitation
bubble dynamics and showed that for higher values of
viscosity the collapse intense becomes less significant.
Tomita et al. (2002) investigated that wall curvature changes
the pattern and collapse rate of cavitation bubbles
significantly and showed that bubble collapse is more
violent in the vicinity of convex rather than flat or concave
boundaries.
Some experimental and numerical works have been
conducted to investigate the effects of physical properties of
fluids on cavitation. Especially surface tension has been
considered in several studies. Kuvshinov (1991) showed
that surface tension has an increasing effect on bubble
surface velocity, and a decreasing effect on minimum
bubble volume of collapsing cavitation bubbles. Zhang &
Zhang (2004) showed that if Buoyancy and Bjerknes forces
are small enough, surface tension will have a significant
role on bubble pattern and jet direction, otherwise, it doesn't
have any effects on bubble dynamics except shortening the
collapse duration and making the jet wider. Liu et al. (2009)
experimentally illustrated higher values of surface tension
results in higher erosion rate, shorter collapse time. Iwai &
Li (2003) changed the surface tension of water by adding
photographic wetting agent and concluded that reducing
surface tension results in less stability of bubble clusters and
decreases the erosion rate.
In this study the effect of surface tension on cavitation
bubble dynamics is studied in both spherical and
nonspherical cases. This effect is considered on growth and
collapse stages separately. Gilmore model is used for
simulating spherical bubble dynamics and finite volume
method in conjunction with VOF method is used for
simulating nonspherical case.
Governing Equations
Governing equations are divided into two categories of
spherical and nonspherical bubble dynamics. For the
spherical case Gilmore model is used with considering mass
and heat transfer, and for nonspherical case NavierStokes
and energy equations in conjunction with VOF method are
used to solve the flow in axisymmetric coordinates.
Spherical bubble dynamics
A spherical bubble containing argon is assumed in quiescent
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
water to interact with an acoustic pressure field. The
Gilmore model is considered as the governing equation of
spherical bubble dynamics in an inviscid and compressible
liquid. The one dimensional equation describing the bubble
dynamics is as follow (Gilmore 1952):
C 2 3C
.*.A2K(1)
Rc Rc[ c)
where R is bubble radius, and C and H are the speed of
sound and liquid enthalpy at the liquidgas interface
respectively. The dots in equation (1) represent the time
derivatives of variables. Sound speed and enthalpy can be
calculated as (Minsier & Proost 2008):
H = n  x
Po1 n 1 P,+B
O K Ol +BnJ (2)
(P+B) n (Pn +B) n
nl
C2 = (PO + B)ln(P +B) (3)
Po
Where po=997.4 kg/m3 is the ambient liquid density,
coefficients n=7.025 and B=3045 bar, come from
experimental results of Holzfuss et al. (1998), P, is the
liquid pressure at infinity taken as P,= Po P 2''
P0O= atm, is the ambient pressure, P,=120 kPa, is the
acoustic wave amplitude, and f=10 kHz, is the derivative
frequency, and P is the liquid pressure at the liquidgas
interface which is calculated as:
2a R
P = Pb  + 4 (4)
R R
where a is surface tension, p is the liquid viscosity, and Pb
is the gas pressure which is calculated by using a van der
Waals type equation of state as follow:
Nto (t)kT
Pb (t) = 4.(t3
4 [(R())3 (R,(t)/8.86)3] (5)
3
where No0(t) is the total number of particles and it is varied
according to condensation and evaporation of water vapor
(Toegel et al. 2000), k1.38x1023 j/mol.K is the Boltzman
constant, T is bubble contents temperature. Total number of
particles can be calculated as Nto(t) NAr+NH2o(t), where NAr,
and NH2o(t) are the number of argon, and water vapor
particles respectively. Ro(t) is the equilibrium radius under
ambient condition which is dependent on time and the total
number of particles (Toegel et al. 2000) as follows:
Po + [(Ro) J] = NokTO (6)
3 R, 8.86
In order to calculate the change of water vapor moles in
bubble, HertzKnudsen model (Holzfuss 2005) is used as
follow to calculate the molar rate of water vapor change:
i evap cond
'H20 H20 H20
4R2 s(T )sat ()] )
M 4 H20 ,H20
H20
where *ho, and mH2o are the molar rate of evaporation,
and condensation of water. a=0.4 is the evaporation
coefficient, c(TS) = 8RgsT, /(rMH2O) is the average
velocity of molecules, PgH2 is the water vapor density in
the bubble, PgH20 =0.0173 kg/m3 is the saturated vapor
density, T,=To is the bubble surface temperature, Rgas=8.314
j/mol.K is the gas constant, and MfH2o is the molecular mass
of water.
For computation of temperature change inside the bubble,
the first law of thermodynamics for an open system is used
as follow (Hatsopoulos & Keenan 1965):
E= hgH20 + 0 W (8)
where E is the rate of internal energy change, h denotes the
enthalpy per water molecule which is given by h=8/2kTo
(Toegel et al. 2000), Q is the heat transferred to the
bubble, and W = PV is the work done by the bubble
which only is the expansion work (Toegel et al. 2000),
where V is the rate of bubble volume change. The
internal energy containing the translational and internal
energy of water molecules, and the translational energy of
argon atoms is
E= 3 N kT + +j eOlT NH2kT (9)
2 2 (e 0 1T \
where 01=5403.4 K, 02=5262.8 K, 03=2294 K are the
characteristic vibrational temperatures (Fay 1965). The heat
transferred to the bubble is
T T
Q= 4 7R2 2 m
]th
t = R Rin ,
th z R *
where lth is the instantaneous diffusive penetration depth,
Z~=mix/(pmixcp,mix) is the thermal diffusivity that is
calculated from pmCpm =4nRk+2.5nAk (Toegel et al. 2000)
where nR corresponds to the equilibrium density at the wall
(Toegel et al. 2000), and Am., is an effective thermal
conductivity of a mixture of polyatomic gases (Hirschfelder
et al. 1954).
With combining the above equations together, the
temperature change inside the bubble can be calculated as
follows (Toegel et al. 2000):
=Q PV
C C
r __ Nk (11)
2 T TT eI
2 2 e B 1 C
where C, is calculated as follow (Toegel et al. 2000):
3 6 + ( 0 I1T
C =Nk + + 7 1T) N NH2,,0k (12)
2 2 [(e 0, \ I )T
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Nonspherical bubble dynamics
In the collapse stage of a bubble near a rigid boundary, flow
is assumed to be laminar, the liquid phase is incompressible,
but the gas phase is assumed to be ideal gas without
considering mass transfer, and the flow is axisymmetric
without swirl. VOF method is used to track the interface
between the liquid and the gas phases. The governing
equations for the flow domain in axisymmetric coordinates
are as follow:
Continuity equation is:
P + V.(/~)= 0 (13)
at
where p is fluid density, and v is the velocity field in the
whole domain. Momentum equations are:
at (14)
VPf + V.[/(V + V ) + pg +
where Pf is pressure domain, g is gravity force vector, and
F is surface tension force which is calculated as
F = (11 where K is the curvature, and n is the unit
normal vector to the interface. Energy equation is:
S(pE)+ V.( Ef +Pf)= V.(kfV, ) (15)
at
Where Ef is the enthalpy of the fluid, Tf is the temperature
field, and kfy is conductivity coefficient of the fluid. As
VOF method is based on one fluid approach, fluid density
and viscosity are calculated according to the following
equations:
S= pF + pg ( F) (16)
I = yF + tg (1 F) (17)
where F is volume fraction of liquid. The ideal gas model is
used for the gas inside the bubble
S= Rgasf (18)
In VOF method at least one of the phases should be
incompressible, and the advection equation of volume
fraction is written for that phase. In the absence of mass
transfer, the advection equation of volume fraction is
8F
+ .VF = 0 (19)
8t
Numerical Scheme
Spherical bubble dynamics
As the governing equations of spherical bubble dynamics
are parabolic ordinary differential equations with respect to
time, forth order RungeKutta method is used to solve the
equations (1), (7), and (11). In order to find the equilibrium
radius under ambient condition Ro, Muller method is used to
solve nonlinear equation (6).
Nonspherical bubble dynamics
Gambit 2.2.30 is used for mesh generation and boundary
type's attribution and the commercially available CFD
code, FL ii i' 6.2 was used to solve the governing
equations. A uniform, quadratic, structured, and 2D grid in
axisymmetric coordinates was generated to simulate the
bubble collapse near a rigid boundary. In order to make the
solution results independent of the boundary conditions,
the solution domain dimensions were set 30 times greater
than the bubble radius. The solution domain was filled by a
coarse 100 x 100 grid, and was adapted 6 times in the
region where the bubble is placed. Boundary conditions at
the near bubble boundaries are axis of symmetry, and wall,
and constant pressure condition was selected for the two
far boundaries. A schematic of solution domain is
illustrated in Fig. 1. In order to ensure the grid
independency, simulations for two similar cases with
different grid resolutions (23.4% difference between the
element numbers) were done, and the resultant differences
between two mesh sizes for bubble volume and maximum
jet velocity are less than 0.5%.
Figure 1: Schematic of the
conditions, and grid generation.
The finite volume method via the pressure based segregated
algorithm was employed to discretize the axisymmetric,
unsteady, and laminar equations (Eqs. 1317). PRESTO
scheme was employed to discretize the pressure equation,
and second order upwind scheme to discretize the
convective and diffusive fluxes in the momentum, and
energy equations. Pressure implicit with splitting of
operators (PISO) algorithm on a collocated grid was
adopted for pressurevelocity coupling. Appropriate
underrelaxation factors were used to stabilize the solution,
and the solution was assumed to converge when the sum of
relative errors is less than 106.
Solution Verification
Spherical bubble dynamics
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
For verifying the accuracy of the chosen model for
simulation of spherical bubble dynamics, present results are
compared with results of Toegel et al. (2000). Figure 2
shows proceeding of bubble radius during the time for a
bubble with initial radius of Ro=4.5 [m at the initial
temperature of T=300 K in contact with a shock wave with
amplitude of P,=1.2 atm and frequency off=26.5 kHz. A
good agreement with the results of Toegel et al. is observed.
30
25
20
.15
P:
t (ps)
Figure 2: Bubble radius in spherical bubble dynamics with
Ro = 4.5 gm, To = 300 K, P, = 1.2 bar,f= 26.5 kHz.
Nonspherical bubble dynamics
In order to confirm the accuracy of the solution
method for modeling Rayleigh collapse near a rigid wall,
three bubbles with the same initial radiuses of Rma=1.45
mm, and different values of standoff as y=1.2, 1.6, and 2.5
have been simulated. Bubbles profiles at three different
moments during the collapse stage are shown in Fig. 3,
which are in a good qualitative agreement with
experimental figures presented in Fig. 3 (ac) of Philipp and
Lauterbom (1998). Because of the extremely high velocity
of bubble surface, measuring maximum jet velocity is
difficult with experimental devices. Therefore, different
values for maximum jet velocity have been declared in
literature. For instance, Benjamin and Ellis (1996) and
Gibson (1968) presented Vma=175, and 160 m/s respectively
for maximum jet velocity, Lauterbom (1974) reported the
value of jet velocities between 100 m/s< v <200 m/s at
normal pressures, and Shima et al. (1981) and Vogel et al.
(1989) calculated Vm,=170 and 156 m/s respectively. Also
results of Fig. 5 of Philipp and Lauterbom (1998) show that
maximum jet velocity increases with increasing standoff
value. Calculated maximum velocities in the present study
are illustrated in table 1, which are in the range of
aforementioned values. Also present results show that
maximum jet velocity increases with increasing standoff
value, as stated by Philipp and Lauterbom(1998). However,
maximum jet velocity for the case of y=2.5 reaches more
than 200 m/s at the last moments of the collapse stage. The
bubble behavior becomes similar to spherical bubble by
increasing standoff parameter. Therefore, because mass
transfer is neglected in the case of nonspherical bubble
dynamics, jet velocity overestimated with respect to the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
surface tension of water in usual conditions. Results show
that jet velocities for the cases with o=0.115 N/m are higher
than those of the cases with o=0.029 N/m. However, the
differences are less than 0.5 % and bubble patterns are
almost similar.
!Ai
b)
Ii
rQ
c)
Figure 3: bubble profiles at three different moments during
the collapse stage with initial radiuses of 1.45 mm, and
standoff parameters of (a) y=1.2, (b) y=1.6, and (c) y=2.5.
Ambient and bubble pressures are P,=1 atm, and Pb=2980
Pa respectively. In comparison with the experimental results
of Philipp and Lauterborn (1998) (with permission of the
authors).
Table 1: Maximum jet velocities of bubbles with initial
radius of R,,m=1.45 mm, and standff values of y=1.2, 1.6,
and 2.5. Ambient and bubble pressures are P,=l atm, and
Pb=2980 Pa respectively.
y 1.2 1.6 2.5
v,, (m/s) 122.17 153.23 183.85
Results and Discussion
Results are divided into nonspherical and spherical cases. In
nonspherical case three bubbles with different radii and
pressure of Pb=3600 Pa are considered in two different
ambient pressures of Pab=100 kPa and 15 kPa. Figure 4
shows the collapse patterns of bubbles with initial radii of a)
Rax=25 [m, b) R,,x=100 im, and c) Rx=300 [m in
ambient pressure of P,,b=100 kPa and surface tension of
o=0.0717 N/m in four different moments. Jet pattern for
smaller bubbles is wider during the time and the length of
jet for larger bubbles is longer. Table 2 contains maximum
jet velocities for two surface tension coefficients of o=0.029
N/m and 0.115 N/m which are smaller and greater than
(a) (b) (c)
Figure 4: Collapse patterns for bubbles with initial radii of
a) Rmax=25 [m, b) R,,x=100 [im, and c) R,,x=300 [m in
ambient pressure of Pa,,b=100 kPa and surface tension of
=0.0717 N/m.
Table 2: Maximum jet velocities of bubbles in ambient
pressure ofPa=l atm, and surface tension coefficients of
=a0.029, 0.115 (N/m). [m/s]
Rma (im) 25 100 300
a=0.029 (N/m) 121.5 121.6 122.2
a=0.115 (N/m) 122.1 122 122.6
Similar simulations have been done in ambient pressure of
Pamb=15 kPa which are shown in figure 5. In this case the
effect of viscosity is more important and causes the bubble
surface not to move apart from the wall. Therefore, the part
of the bubble surface which is close to wall is flatter. This
effect is more important for smaller bubbles. Maximum jet
velocities for two surface tension coefficients of o=0.029
N/m and 0.115 N/m are shown in table 3. Results show that
the differences between maximum jet velocities are greater
than those of the ambient pressure of 100 kPa. However, the
differences are less than 1 %.
(a) (b) (c)
Figure 5: Collapse patterns for bubbles with initial radii of
a) R,,,=25 [im, b) Rx=100 gim, and c) Rx=300 gm in
ambient pressure of Pab=15 kPa and surface tension of
o=0.0717 N/m.
Table 3: Maximum jet velocities of bubbles in ambient
pressure ofPa=15 kPa, and surface tension coefficients of
=a0.029, 0.115 (N/m). [m/s]
Rma (gm) 25 100 300
a00.029 (N/m) 21.93 27 27.92
a00.115 (N/m) 22.10 27.16 27.97
All together, it can be inferred that surface tension does not
have a significant role in the dynamics of collapsing
bubbles near a rigid boundary.
In order to evaluate surface tension role in growth stage,
experiments.
a)
U
F7 7 T i I t
Q
6 )
6L 6L
growth and collapse stages of spherical bubble dynamics are
considered in a sinusoidal acoustic pressure field.
Simulations have been done for three different situations of
surface tension coefficient as o=0, 0.0717, and 0.115 N/m.
Figure 6 shows radius and velocity changes versus time of a
bubble with initial radius of Ro=5 [m in a pressure field
with frequency of f=25 kHz and pressure amplitude of
P,=120 kPa. Results show that increasing surface tension
causes the maximum radius and the growth time to decrease.
Therefore, the collapse stage for the cases with larger
surface tension coefficients is less violent and total collapse
duration is shorter.
40
/ 0=o0 N/m
35 / 0 0717N/fm
o=0115 N/m
30
25
220
15
10
5
0 5 10 15 20 25 30 35 40
t (G s)
500
o=0N/m
o=0 0717 N/m
 o=0 115 N/m
500
1000
0 10 20 30 40
t (G S)
Figure 6: Radius and velocity versus time of bubbles with
initial radii of Ro=5 [m,f=25 kHz, P,=120 kPa and surface
tension coefficients of a=0, 0.0717, and 0.115 N/m.
For understanding surface tension effects in different
frequencies, similar simulations as those of figure 6 but in a
pressure field with frequency of f=5 kHz have been
performed and are shown in figure 7. In this case, increasing
surface tension decreases the collapse rate and duration, but
the differences between maximum bubble radius, maximum
velocity and collapse time are less than those of the case
with f=25 kHz. Because in lower frequency pressure field
growth duration is longer and let the bubble become larger.
Therefore, the effect of surface tension becomes less
significant.
Figure 8 shows the results of the simulations of bubble
dynamics with initial radius of Ro=5 gm in a pressure field
with frequency of f=25 kHz and pressure amplitude of
P,=150 kPa. In this case the decrement of velocity and
collapse duration by increasing surface tension is much less
than those of with lower pressure amplitude. Because in this
situation the ratio of surface tension force and the force
caused by pressure difference between bubble content and
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
ambient pressure is smaller than that of the case with lower
pressure amplitude.
0 100 200
t (j s)
1000
500
500
2 1000
> 1500
2000
 =0 N/m
2500
500 = 00717N/m
3000 0=0115N/m
3500
0 100 200
t (L s)
Figure 7: Radius and velocity versus time of bubbles with
initial radii of Ro=5 gm, f=5 kHz, P,=120 kPa and surface
tension coefficients of o=0, 0.0717, and 0.115 N/m.
0 5 10 15 20 25 30 35 40
t (G s)
1000
o = 0 NIm
500o 0o= 0717N/m
 G=0 115N/m
S500
> 1000
1500
2000
2500
0 10 20 30 40
t (g s)
Figure 8: Radius and velocity versus time of bubbles with
initial radii of Ro=5 gm, f=25 kHz, P,=150 kPa and surface
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
tension coefficients of o=0, 0.0717, and 0.115 N/m.
Figure 9 shows similar results as those of figure 8 but in a
pressure field with frequency off=5 kHz. Results show that
as a result of decrement in pressure frequency, the
differences between velocity and collapse duration are
decreased. Comparison of the results of figure 6 and figure
9 show that surface tension effects on collapse rate of
cavitation bubbles in pressure fields with higher frequencies
and lower pressure amplitudes are significant, but
decreasing frequency and increasing pressure amplitude
reduces surface tension effects on cavitation bubble
dynamics.
i 200
C4 1^n
2000
0
2000
4000
o=ON/m
6000 c=0 0717N/m
 c=0115N/m
8000
0 100 200
t (gL s)
Figure 9: Radius and velocity versus time of bubbles with
initial radii of Ro=5 im, f=5 kHz, P,=150 kPa and surface
tension coefficients of o=0, 0.0717, and 0.115 N/m.
To have a general conclusion about surface tension effects
on cavitation bubbles, bubble dynamics in a wide range of
surface tension coefficient, bubble initial radius, and
pressure frequency is simulated and the results are shown in
figures 10 and 11. Figure 10a shows that maximum
velocity reduces significantly with increasing surface
tension, especially for smaller bubbles. But figures 10b,c
show that with increasing pressure amplitude, the gradient
of velocity with respect to surface tension decreases.
However, in high frequencies surface tension is important
for smaller bubbles. Results of figure 11 show that in this
case, the slope of velocity diagram with respect to surface
tension is smaller than that of the case with higher pressure
frequencies. For higher values of pressure amplitude, the
gradients of velocity diagrams with respect to surface
tension, which are shown in figure 11c, become smaller
and show that in low pressure frequencies and high pressure
amplitudes surface tension force is much smaller than
inertial forces.
1500
1000
500
(b)
1500
RO =4 pm
RO=7pm
RO =10 Pm
  
S 002 004 006 008 0.1 0
S(N/m)
RO =4 Im
 RO=7 m
 = 10
1000
JUUU I I I i
2000
> 1500 .
1000
0 0.02 0.04 0.06 0.08 01 0.12
a (N/m)
Figure 10: Maximum bubble velocity versus surface
tension in conditions off=25 kHz, a) Pe=1.2 bar; b) Pe=1.3
bar; c)Pe=1.5 bar; Ro=4, 7, and 10 pm.
S3000
2000
14000
3000
2000
10000
C)
8000
6000
4000
S 002 0.04 006 008 01 01:
S(N/m)
RO =4 pm
R =7 Tpm 
RO = 10 Pm
0 0.02 0.04 006 0.08 0.1 0.1:
o (N/m)
 RD=7 pm
RO RD= 10 pm
0 0 02 0 04 0 06 008 0 1 0.1
o(N/m)
Figure 11: Maximum bubble velocity versus surface
tension in conditions off=5 kHz, a) Pe=1.2 bar; b) Pe=1.3
bar; c)Pe=1.5 bar; Ro=4, 7, and 10 gm.
0 002 004 006
S(N/m)
RO =4 pm
 ^^ R0 = 10 #rn
RO =4 pm
RO=7
    R= lO pm
008
0.1 012
2
Conclusions
In the present study we have studied surface tension effects
on the dynamics of spherical and nonspherical cavitation
bubbles in different conditions of bubble content and
ambient liquid. Gilmore model is used for simulation of
spherical bubble dynamics and NavierStokes and energy
equations in conjunction with VOF method are used to
simulate bubble collapse adjacent to a rigid wall. Results
show that in the whole range of pressure difference between
bubble content and ambient liquid, and initial bubble radius,
surface tension does not have any important roles in
collapse stage of cavitation bubbles. But, simulation of
growth and collapse stages of spherical bubble dynamics in
an acoustic pressure field shows that surface tension has
important effects on growth stage, and results in more
violent collapse stages. However, this effect becomes less
significant in high values of pressure amplitude and lower
values of pressure frequency.
References
Benjamin, T.B. & Ellis, A.T. The collapse of cavitation
bubbles and the pressure thereby produced against solid
boundaries. Phil. Trans. R. Soc. Lond. A Vol. 260, 221240
(1966)
Brennen, C.E. Cavitation and Bubble Dynamics. Oxford
University Press (1995)
Fay, J.A. Molecular Thermodynamics. AddisonWesley,
Reading, MA (1965)
Gibson, D.C. Cavitation adjacent to plane boundaries. In
Proc. Third Australasian Conference on Hydraulics and
Fluid Mechanics. The Australian Institution of Engineers
(1968)
Gilmore, F.R. Hydrodynamics laboratory report 26.4,
California Institute of Technology (1952)
Hatsopoulos, G.N. & Keenan, J.H. Principles of General
Thermodynamics. Wiley, New York, (1965)
Hirschfelder, J.O., Curtiss, C.F. & Bird, R.B. Molecular
Theory of Gases and Liquids, Wiley, New York (1954)
Holzfuss, J. Unstable diffusion and chemical dissociation of
a single sonoluminescing bubble. Phys. Rev. E Vol. 71,
026304 (2005)
Holzfuss, J., RUggeberg, M. & Billo, A. Shock Wave
Emissions of a Sonoluminescing Bubble. Phys. Rev. Lett.
Vol. 81, 54345437 (1998)
Iwai, Y & Li, S. Cavitation erosion in waters having
different surface tensions. Wear Vol. 254, 19 (2003)
Kamath, VA., Prosperetti, A. and Egolfopoulos, F.N. A
theoretical study of sonoluminescence. J. Acoust. Soc. Am.
Vol. 94, 248260 (1993)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Kuvshinov, G.I. Effect of surface tension on the collapse of
a cavitation bubble. J. Eng. Phys. Thermophys. Vol. 60
3437 (1991)
Lauterbom, W. Kavitation durch Laserlicht. Acustica Vol.
31, 5158 (1974)
Lindau, O. and Lauterbom, W, Cinematographic
observation of the collapse and rebound of a laserproduced
cavitation bubble near a wall. J. Fluid Mech. vol. 479,
327348(2003)
Liu, X.M., He, J., Lu, J. & Ni, X.W. Effect of surface
tension on a liquidjet produced by the collapse of a
laserinduced bubble against a rigid boundary. Optics &
Laser Technol. Vol. 41, 2124 (2009)
Minsier, V & Proost, J. Shock wave emission upon
spherical bubble collapse during cavitationinduced
megasonic surface cleaning. Ultrasonics Sonochem. Vol. 15,
598604 (2008)
Naude, C.F. & Ellis A.T. On the mechanism of cavitation
damage by nonhemispherical cavities in contact with a
solid boundary. ASME J. Basic Eng. Vol. 83, 648656
(1961)
Philipp A. and Lauterbom W. Cavitation erosion by single
laserproduced bubbles. J. Fluid Mech. vol. 361, Issue 01,
75116 (1998)
Plesset, M.S. The dynamics of cavitation bubbles. ASME J.
Appl. Mech., Vol. 16, 228231 (1949)
Plesset, M.S. & Prosperetti, A. Bubble dynamics and
cavitation. Annu. Rev. Fluid Mech. Vol. 9, 145185 (1997)
Plesset, M.S. & Zwick, S.A. A nonsteady heat diffusion
problem with spherical symmetry. J. Appl. Phys. Vol. 23,
No. 1,9598 (1952)
Popinet S. and Zaleski S. Bubble collapse near a solid
boundary: a numerical study of the influence of viscosity. J.
Fluid. Mech. Vol. 464, 137163 (2002)
Prosperetti, A. Thermal effects and damping mechanism in
the forced radial oscillations of gas bubbles in liquid. J.
Acoust. Soc., Vol. 61, 1727 (1977)
Rayleigh, L. Pressure due to collapse of bubbles. Philos.
Mag. Vol. 34, 9498 (1917)
Reynolds, O. The causes of the racing of the engines of
screw steamers investigated theoretically and by experiment.
Trans. Inst. Naval Arch., Vol. 14, 5667 (1873)
Shima, A., Takayama, K., Tomita, Y & Miura, N. An
experimental study on effects of a solid wall on the motion
of bubbles and shock waves in bubble collapse. Acustica
Vol. 48, 293301 (1981)
Toegel R., Gompf B., Pecha R. and Lohse D. Dose Water
Vapor Prevent Upscaling Sonoluminescence?. Phys. Rev.
Lett. Vol. 85, No. 15, 31653168(2000)
Tomita, Y, Robinson, PB., Tong, R.P and Blake, J.R.
Growth and collapse of cavitation bubbles near a curved
rigid boundary J. Fluid Mech., vol. 466, 259283 (2002)
Vogel,A., Lauterbom, W. & Timm, R. Optical and acoustic
investigations of the dynamics of laserproduced cavitation
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
bubbles near a solid boundary. J. Fluid Mech. Vol. 206,
299338 (1989)
Zhang, Z. & Zhang, H. Surface tension effects on the
behavior of a cavity growing, collapsing, and rebounding
near a rigid wall. Phys. Rev. E Vol. 70 056310 (2004)
